<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="dtconstr.html#K18" title="DTCONSTR:func.18">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<a NAME="E1:56"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
( <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> or <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="dtconstr.html#E15"><span class="lab"><font color="Green" title="E10">Lm2</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dtconstr.html#K18" title="DTCONSTR:func.18">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K18" title="DTCONSTR:func.18">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:56_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K18" title="DTCONSTR:func.18">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><a NAME="E2:56_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">tsg</font> being   <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">TerminalString</a> <font color="Olive" title="b1">tsg</font> &amp; <font color="Olive" title="b1">tsg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  &amp; <font color="Olive" title="b1">tsg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">nt</font> )
 ;<br/><a NAME="E3:56_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">O</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dtconstr.html#T14" target="_self" title="DTCONSTR:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:56_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K18" title="DTCONSTR:func.18">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:56"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K18" title="DTCONSTR:func.18">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font></span><br/>

<a NAME="E4:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">O</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">prtO</font> =  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c6">O</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">rtO</font> =  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c6">O</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">srtO</font> = <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">prtO</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  ;<br/>
<a NAME="E8:56"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<font color="Maroon" title="c11">rtO</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></span>;<br/>
<a NAME="E9:56"/><span class="kw">then </span>
<font color="Maroon" title="c7">S</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c11">rtO</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="dtconstr.html#E19"><span class="lab"><font color="Green" title="E13">Lm5</font></span></a>, <a class="ref" href="dtconstr.html#T4" target="_self" title="DTCONSTR:th.4">Th4</a></span>;<br/>
<a NAME="E10:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c7">S</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">prtO</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dtconstr.html#D4" target="_self" title="DTCONSTR:def.4">Def4</a></span>;<br/>
<a NAME="E11:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A5</font></span>: 
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">TerminalString</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c12">srtO</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="dtconstr.html#T14" target="_self" title="DTCONSTR:th.14">Th14</a></span>;<br/>
<a NAME="E12:56"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c11">rtO</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></span>;<br/>
<a NAME="E13:56"/>
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">TerminalString</a> <font color="Maroon" title="c11">rtO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="dtconstr.html#T14" target="_self" title="DTCONSTR:th.14">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:56"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K18" title="DTCONSTR:func.18">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E10:56"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E11:56"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E12:56"><span class="lab"><font color="Green" title="E45">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
